Exercise 4D

What is the nature of the roots of the quadratic equation, $5x(x-2)+6=0$? $($No real $/$Real and unequal $/$Real and equal $)$

Find the value of $k$ for which the equation $x^2+k(2x+k-1)+2=0$ has real and equal roots.

Find the values of $p$ for which the quadratic equation $(p+1)x^2-6(p+1)x+3(p+9)=0, p\ne -1$  has equal roots. Hence, find the roots of the equation. 

Find the values of $k$ for which the given quadratic equation has real and distinct root:
$5x^2-kx+1=0$

If $a$ and $b$ are real and $a\ne b$ then show that the roots of the equation $(a-b)x^2+5(a+b)x-2(a-b)=0$ are real and unequal.

If the roots of the equation $\left(a^2+b^2\right)x^2-2(ac+bd)x+\left(c^2+d^2\right)=0$ are equal, prove that $\frac{a}{b}=\frac{c}{d}$

If $ad\mathit{\ne }bc$ then prove that the equation $\left(a^2+b^2\right)x^2+2(ac+bd)x+\left(c^2+d^2\right)=0$ has no real roots.

If the roots of the equations $a{x}^2+2bx+c=0$ and $b{x}^2-2\sqrt{ac}x+b=0$ are simultaneously real then prove that $b^2=ac$